Inelastic scattering in a local polaron model with quadratic coupling to bosons

We calculate the inelastic scattering probabilities in the wide band limit of a local polaron model with quadratic coupling to bosons. The central object is a two-particle Green’s function which is calculated exactly using a purely algebraic approach. Compared with the usual linear interaction term a quadratic interaction term gives higher probabilities for inelastic scattering involving a large number of bosons. As an application we consider the problem hot-electron-mediated energy transfer at surfaces and use the delta self-consistent field extension of density-functional theory to calculate and compare coupling parameters and probabilities for exciting different vibrational modes of CO adsorbed on a Cu(100) surface.

Figure 1

A general example of adiabatic potentials $V_1\left(x\right)$ and $V_0\left(x\right)$ and the coupling function ${\epsilon }_a\left(a\right)=V_1\left(x\right)-V_0\left(x\right)$. The vertical distance between the two potentials at the ground-state minimum is ${\epsilon }_0$.

Figure 2

A shifted excited state corresponds to a linear coupling function ${\epsilon }_a\left(x\right)\sim x$. The strength of the coupling is proportional to the derivative of the excited state at $x_0$.

Figure 3

A frequency shifted excited state gives rise to a quadratic coupling function: ${\epsilon }_a\left(x\right)\sim x^2$.

Figure 4

Probabilities of making the transition $0\to n$ through resonant inelastic scattering with linear and quadratic coupling. The parameters are $\Gamma /\hslash {\omega }_0=0.5$, $g_1=0.2$, and ${\omega }_1=0.75{\omega }_0$ $\left(g_2=0.02\right)$. Even though $g_2<g_1^2$ the quadratic coupling becomes dominating for large $n$ due to the slowly decaying expansion coefficients. One should also note the spacing between peaks which is ${\omega }_0$ for linear coupling and $2{\omega }_1$ for quadratic coupling. The centers of the probability distributions are approximately shifted by $n{\omega }_0/2$ for the linear coupling and $n{\omega }_1/2$ for the quadratic coupling relative to the bare resonance energy ${\epsilon }_0$, since this is where the binomial coefficients in Eqs. (14, 19) have their maxima.

Figure 5

Potential energy surfaces along the frustrated rotation mode of CO adsorbed on a Cu(100) surface. The coordinate $x$ is a generalized coordinate representing the deviation from equilibrium. $x=0.4$ corresponds to a $24°$ angular deviation from the perpendicular position.

Figure 6

Probabilities of exciting two and four quanta of vibrations to the center-of-mass, internal stretch and frustrated rotation modes CO adsorbed on Cu(100).